Differential equations MOC

Reduction of order (homogenous second-order differential equation)

Reduction of order is a technique for finding the general solution of a homogenous second order linear DE¹ when a particular solution is known.

We begin by assuming that for some function to be determined. It follows from this that

Given that is indeed a solution, this substitution will reduce the DE to a first order separable DE on the independent variable (see reasoning below), which can be then used to determine the general solution.

Explanation

We write the DE as , where the linear operation is defined by

We are given that . Then

where is the second order linear operator

Method

In general, I have found it most effective to only substitute the particular solution after the gathering different terms. Consider the ODE²

with particular solution , so

We assume is a solution to the ODE, thus

which is a separable ODE of order 2.

Practice problems

• 2017. Elementary differential equations and boundary value problems, p. 133 (§3.4 problems 18–22)

---

tidy | en | sembr | review

Footnotes

1. Perhaps generalisable to higher orders when solutions are given? ↩

2. 2017. Elementary differential equations and boundary value problems, p. 133 (§3.4 problem 22) ↩